Of course they can. If they couldn't there would be no point to having a V1 would there.
Just because I'm above max landing weight, it does not mean I can't land. Varying levels of maintenance action will be required...but a rule of thumb is that any runway you can take off from...you can immediately land on.
I may have to dump a lot of fuel. I don't necessarily have to. The circumstances of the day will control that. If I had a problem out of Melbourne, and the weather there was not conducive to a return, I may well use the transit to Sydney to sort some of these issues out.
When you write about an engine vibration for the scenario, you have the luxury of knowing that it is an engine vibration. In the aircraft, it may not be all that obvious. Taking off, a very likely cause of vibration is a tyre failure. It's always best not to jump to conclusions.
